Abstract We present sound and complete sequent calculi for the modal mu-calculus with converse modalities, aka two-way modal mu-calculus. Notably, we introduce a cyclic proof system wherein proofs can be represented as finite trees with back-edges, i.e., finite graphs. The sequent calculi incorporate ordinal annotations and structural rules for managing them. Soundness is proved with relative ease as is the case for the modal mu-calculus with explicit ordinals. The main ingredients in the proof of completeness are isolating a class of non-wellfounded proofs with sequents of bounded size, called slim proofs, and a counter-model construction that shows slimness suffices to capture all validities. Slim proofs are further transformed into cyclic proofs by means of re-assigning ordinal annotations.

We study the interaction of views, queries, and background knowledge in the form of existential rules. The motivating questions concern monotonic determinacy of a query using views w.r.t. rules, which refers to the ability to recover the query answer from the views via a monotone function. We study the decidability of monotonic determinacy, and compare with variations that require the “recovery function” to be in a well-known monotone query language, such as conjunctive queries or Datalog. Surprisingly, we find that even in the presence of basic existential rules, the borderline between well-behaved and badly-behaved answerability differs radically from the unconstrained case. In order to understand this boundary, we require new results concerning entailment problems involving views and rules.

A seminal theorem by van Benthem characterises the bisimulation-invariant fragment of First Order Logic (FOL) in terms of Modal Logic. Similarly, Janin and Walukiewicz have shown that the bisimulation-invariant fragment of Monadic Second Order Logic (MSO) corresponds precisely to the (modal) mu-calculus. Notably, neither argument immediately carries over when only considering finite structures. In the case of FOL, Rosen provided an alternative proof, which also works over finite structures. However, such characterisations for any logic more expressive than FOL over finite structures have remained long-standing open problems. In this paper, we prove such a characterisation for parameter-free Monadic Least Fixpoint Logic, a well-known logic in between FOL and MSO. In particular, we show that, over finite structures, every two-way bisimulation-invariant formula in this logic is equivalent to a formula in the two-way mu-calculus and vice versa.

Abstract Automata operating on infinite objects feature prominently in the theory of the modal $$\mu $$ -calculus. One such application concerns the tableau games introduced by Niwiński & Walukiewicz, of which the winning condition for infinite plays can be naturally checked by a nondeterministic parity stream automaton. Inspired by work of Jungteerapanich and Stirling we show how determinization constructions of this automaton may be used to directly obtain proof systems for the $$\mu $$ -calculus. More concretely, we introduce a binary tree construction for determinizing nondeterministic parity stream automata. Using this construction we define the annotated cyclic proof system $$\textsf{BT}$$ , where formulas are annotated by tuples of binary strings. Soundness and Completeness of this system follow almost immediately from the correctness of the determinization method.

Many algorithmic results on the modal mu-calculus use representations of formulas such as alternating tree automata or hierarchical equation systems. At closer inspection, these results are not always optimal, since the exact relation between the formula and its representation is not clearly understood. In particular, there has been confusion about the definition of the fundamental notion of the size of a mu-calculus formula. We propose the notion of a parity formula as a natural way of representing a mu-calculus formula, and as a yardstick for measuring its complexity. We discuss the close connection of this concept with alternating tree automata, hierarchical equation systems and parity games. We show that well-known size measures for mu-calculus formulas correspond to a parity formula representation of the formula using its syntax tree, subformula graph or closure graph, respectively. Building on work by Bruse, Friedmann & Lange we argue that for optimal complexity results one needs to work with the closure graph, and thus define the size of a formula in terms of its Fischer-Ladner closure. As a new observation, we show that the common assumption of a formula being clean, that is, with every variable bound in at most one subformula, incurs an exponential blow-up of the size of the closure. To realise the optimal upper complexity bound of model checking for all formulas, our main result is to provide a construction of a parity formula that (a) is based on the closure graph of a given formula, (b) preserves the alternation-depth but (c) does not assume the input formula to be clean.

Algorithms for solving computational problems related to the modal μ-calculus generally do not take the formulas themselves as input, but operate on some kind of representation of formulas. This representation is usually based on a graph structure that one may associate with a μ-calculus formula. Recent work by Kupke, Marti & Venema showed that the operation of renaming bound variables may incur an exponential blow-up of the size of such a graph representation. Their example revealed the undesirable situation that standard constructions, on which algorithms for model checking and satisfiability depend, are sensitive to the specific choice of bound variables used in a formula.

We study complete approximations of an ontology formulated in a non-Horn description logic (DL) such as ALC in a Horn DL such as EL. We provide concrete approximation schemes that are necessarily infinite and observe that in the ELU-to-EL case finite approximations tend to exist in practice and are guaranteed to exist when the source ontology is acyclic. In contrast, neither of this is the case for ELU_bot-to-EL_bot and for ALC-to-EL_bot approximations. We also define a notion of approximation tailored towards ontology-mediated querying, connect it to subsumption-based approximations, and identify a case where finite approximations are guaranteed to exist.

Game logic was introduced by Rohit Parikh in the 1980s as a generalisation of propositional dynamic logic (PDL) for reasoning about outcomes that players can force in determined 2-player games. Semantically, the generalisation from programs to games is mirrored by moving from Kripke models to monotone neighbourhood models. Parikh proposed a natural PDL-style Hilbert system which was easily proved to be sound, but its completeness has thus far remained an open problem. In this paper, we introduce a cut-free sequent calculus for game logic, and two cut-free sequent calculi that manipulate annotated formulas, one for game logic and one for the monotone μ -calculus, the variant of the polymodal μ -calculus where the semantics is given by monotone neighbourhood models instead of Kripke structures. We show these systems are sound and complete, and that completeness of Parikh's axiomatization follows. Our approach builds on recent ideas and results by Afshari & Leigh (LICS 2017) in that we obtain completeness via a sequence of proof transformations between the systems. A crucial ingredient is a validity-preserving translation from game logic to the monotone μ -calculus.

We use the connection between automata and logic to prove that a wide class of coalgebraic fixpoint logics enjoys uniform interpolation. To this aim, first we generalize one of the central results in coalgebraic automata theory, namely closure under projection, which is known to hold for weak-pullback preserving functors, to a more general class of functors, i.e., functors with quasifunctorial lax extensions. Then we will show that closure under projection implies definability of the bisimulation quantifier in the language of coalgebraic fixpoint logic, and finally we prove the uniform interpolation theorem.